The Limit of a Function

1. Determine the limit of the following nine problems. solution

$\displaystyle \lim_{x \to 1 } \, \dfrac{x^2-1}{x-1}$   $\displaystyle \lim_{x \to -1 } \, \dfrac{x^2-1}{x+1}$  $\displaystyle \lim_{x \to 2 } \, \dfrac{x^2-4}{x-2}$  $\displaystyle \lim_{x \to 1 } \, \dfrac{x^2-3x+2}{2x-2}$

$\displaystyle \lim_{x \to -2} \, \dfrac{x^2+5x+6}{x^2+2x}$   $\displaystyle \lim_{x \to 3 } \, \dfrac{x^2-5x+6}{x^2-9}$   $\displaystyle \lim_{x \to \infty} \, \dfrac{2x^3-6x^2+5}{6x^3-3x+5}$

$\displaystyle \lim_{x \to \infty} \, \dfrac{2x^5-6x^2+5}{3x^4-3x+5}$   $\displaystyle \lim_{x \to- \infty} \, \dfrac{5x^4-6x+15}{2x^5-x^2+7}$

2. Determine the limit of the following five problems.  solution

$\displaystyle \lim_{x \to \frac{1}{2}}\cos (\pi x) -e^{x-\frac{1}{2}}$ $\displaystyle \lim_{x \to -1}\frac{x^2-x-2}{7x+7}$ $\displaystyle \lim_{h \to 0}\frac{\sqrt{16+h}-4}{h}$

$\displaystyle \lim_{x \to 0}\dfrac{\frac{1}{x+4}-\frac{1}{4}}{x}$ $\displaystyle \lim_{x \to \infty} \, \dfrac{3x^5+5x+2}{2x^5-2x^3+1}$